Page 76 - Applied Statistics And Probability For Engineers

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2-1.4 Counting Techniques (CD Only)

In many of the examples in Chapter 2, it is easy to determine the number of outcomes in each
event. In more complicated examples, determining the number of outcomes that comprise the
sample space (or an event) becomes more difficult. To associate probabilities with events, it is
important to know the number of outcomes both in an event and in the sample space. Some
simple rules can be used to simplify the calculations.
In Example 2-4, an automobile manufacturer provides vehicles equipped with selected
options. Each vehicle is ordered
With or without an automatic transmission
With or without air conditioning
With one of three choices of a stereo system
With one of four exterior colors
The tree diagram in Fig. 2-6 describes the sample space of all possible vehicle types. The size
of the sample space equals the number of branches in the last level of the tree and this quantity
equals 2 2 3 4 = 48. This leads to the following useful result.

Multiplication
Rule (for If an operation can be described as a sequence of k steps, and
counting
if the number of ways of completing step 1 is n , and
techniques) 1
if the number of ways of completing step 2 is n for each way of completing
2
step 1, and
if the number of ways of completing step 3 is n for each way of completing
3
step 2, and so forth,
the total number of ways of completing the operation is

n n p n k
2
1

EXAMPLE S2-1 In the design of a casing for a gear housing, we can use four different types of fasteners,
three different bolt lengths, and three different bolt locations. From the multiplication rule,
4 3 3 36 different designs are possible.

Permutations
Another useful calculation is the number of ordered sequences of the elements of a set.
Consider a set of elements, such as S {a, b, c}. A permutation of the elements is an ordered
sequence of the elements. For example, abc, acb, bac, bca, cab, and cba are all of the permu-
tations of the elements of S.

The number of permutations of n different elements is n! where
n! n 1n 12 1n 22 p 2 1 (S2-1)

2-1